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# Polytope of Type {18,18}

Atlas Canonical Name : {18,18}*1944n
if this polytope has a name.
Group : SmallGroup(1944,946)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,9}*972e
3-fold quotients : {6,18}*648e
6-fold quotients : {6,9}*324d
9-fold quotients : {6,6}*216a
18-fold quotients : {6,3}*108
27-fold quotients : {6,6}*72b
54-fold quotients : {6,3}*36
81-fold quotients : {2,6}*24
162-fold quotients : {2,3}*12
243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,167)( 83,168)( 84,166)( 85,165)
( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)( 93,184)
( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)(101,177)
(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)(109,194)
(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)(117,198)
(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)(125,215)
(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)(133,205)
(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)(141,218)
(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)(149,235)
(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)(157,228)
(158,226)(159,227)(160,232)(161,233)(162,234)(247,251)(248,252)(249,250)
(253,262)(254,263)(255,264)(256,269)(257,270)(258,268)(259,267)(260,265)
(261,266)(274,278)(275,279)(276,277)(280,289)(281,290)(282,291)(283,296)
(284,297)(285,295)(286,294)(287,292)(288,293)(301,305)(302,306)(303,304)
(307,316)(308,317)(309,318)(310,323)(311,324)(312,322)(313,321)(314,319)
(315,320)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)(331,412)
(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)(339,425)
(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)(347,415)
(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)(355,435)
(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)(363,454)
(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)(371,447)
(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)(379,464)
(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)(387,468)
(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)(395,485)
(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)(403,475)
(404,476)(405,477);;
s1 := (  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)
(  9, 86)( 10, 93)( 11, 92)( 12, 91)( 13, 99)( 14, 98)( 15, 97)( 16, 96)
( 17, 95)( 18, 94)( 19,101)( 20,100)( 21,102)( 22,107)( 23,106)( 24,108)
( 25,104)( 26,103)( 27,105)( 28,162)( 29,161)( 30,160)( 31,159)( 32,158)
( 33,157)( 34,156)( 35,155)( 36,154)( 37,143)( 38,142)( 39,144)( 40,140)
( 41,139)( 42,141)( 43,137)( 44,136)( 45,138)( 46,151)( 47,153)( 48,152)
( 49,148)( 50,150)( 51,149)( 52,145)( 53,147)( 54,146)( 55,125)( 56,124)
( 57,126)( 58,122)( 59,121)( 60,123)( 61,119)( 62,118)( 63,120)( 64,133)
( 65,135)( 66,134)( 67,130)( 68,132)( 69,131)( 70,127)( 71,129)( 72,128)
( 73,117)( 74,116)( 75,115)( 76,114)( 77,113)( 78,112)( 79,111)( 80,110)
( 81,109)(163,167)(164,166)(165,168)(169,170)(172,175)(173,177)(174,176)
(179,180)(181,186)(182,185)(183,184)(187,189)(190,235)(191,237)(192,236)
(193,241)(194,243)(195,242)(196,238)(197,240)(198,239)(199,219)(200,218)
(201,217)(202,225)(203,224)(204,223)(205,222)(206,221)(207,220)(208,227)
(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)(216,231)
(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)(251,330)
(252,329)(253,336)(254,335)(255,334)(256,342)(257,341)(258,340)(259,339)
(260,338)(261,337)(262,344)(263,343)(264,345)(265,350)(266,349)(267,351)
(268,347)(269,346)(270,348)(271,405)(272,404)(273,403)(274,402)(275,401)
(276,400)(277,399)(278,398)(279,397)(280,386)(281,385)(282,387)(283,383)
(284,382)(285,384)(286,380)(287,379)(288,381)(289,394)(290,396)(291,395)
(292,391)(293,393)(294,392)(295,388)(296,390)(297,389)(298,368)(299,367)
(300,369)(301,365)(302,364)(303,366)(304,362)(305,361)(306,363)(307,376)
(308,378)(309,377)(310,373)(311,375)(312,374)(313,370)(314,372)(315,371)
(316,360)(317,359)(318,358)(319,357)(320,356)(321,355)(322,354)(323,353)
(324,352)(406,410)(407,409)(408,411)(412,413)(415,418)(416,420)(417,419)
(422,423)(424,429)(425,428)(426,427)(430,432)(433,478)(434,480)(435,479)
(436,484)(437,486)(438,485)(439,481)(440,483)(441,482)(442,462)(443,461)
(444,460)(445,468)(446,467)(447,466)(448,465)(449,464)(450,463)(451,470)
(452,469)(453,471)(454,476)(455,475)(456,477)(457,473)(458,472)(459,474);;
s2 := (  1,271)(  2,273)(  3,272)(  4,277)(  5,279)(  6,278)(  7,274)(  8,276)
(  9,275)( 10,284)( 11,283)( 12,285)( 13,281)( 14,280)( 15,282)( 16,287)
( 17,286)( 18,288)( 19,297)( 20,296)( 21,295)( 22,294)( 23,293)( 24,292)
( 25,291)( 26,290)( 27,289)( 28,244)( 29,246)( 30,245)( 31,250)( 32,252)
( 33,251)( 34,247)( 35,249)( 36,248)( 37,257)( 38,256)( 39,258)( 40,254)
( 41,253)( 42,255)( 43,260)( 44,259)( 45,261)( 46,270)( 47,269)( 48,268)
( 49,267)( 50,266)( 51,265)( 52,264)( 53,263)( 54,262)( 55,299)( 56,298)
( 57,300)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)( 64,312)
( 65,311)( 66,310)( 67,309)( 68,308)( 69,307)( 70,315)( 71,314)( 72,313)
( 73,322)( 74,324)( 75,323)( 76,319)( 77,321)( 78,320)( 79,316)( 80,318)
( 81,317)( 82,437)( 83,436)( 84,438)( 85,434)( 86,433)( 87,435)( 88,440)
( 89,439)( 90,441)( 91,450)( 92,449)( 93,448)( 94,447)( 95,446)( 96,445)
( 97,444)( 98,443)( 99,442)(100,451)(101,453)(102,452)(103,457)(104,459)
(105,458)(106,454)(107,456)(108,455)(109,410)(110,409)(111,411)(112,407)
(113,406)(114,408)(115,413)(116,412)(117,414)(118,423)(119,422)(120,421)
(121,420)(122,419)(123,418)(124,417)(125,416)(126,415)(127,424)(128,426)
(129,425)(130,430)(131,432)(132,431)(133,427)(134,429)(135,428)(136,465)
(137,464)(138,463)(139,462)(140,461)(141,460)(142,468)(143,467)(144,466)
(145,475)(146,477)(147,476)(148,472)(149,474)(150,473)(151,469)(152,471)
(153,470)(154,479)(155,478)(156,480)(157,485)(158,484)(159,486)(160,482)
(161,481)(162,483)(163,356)(164,355)(165,357)(166,353)(167,352)(168,354)
(169,359)(170,358)(171,360)(172,369)(173,368)(174,367)(175,366)(176,365)
(177,364)(178,363)(179,362)(180,361)(181,370)(182,372)(183,371)(184,376)
(185,378)(186,377)(187,373)(188,375)(189,374)(190,329)(191,328)(192,330)
(193,326)(194,325)(195,327)(196,332)(197,331)(198,333)(199,342)(200,341)
(201,340)(202,339)(203,338)(204,337)(205,336)(206,335)(207,334)(208,343)
(209,345)(210,344)(211,349)(212,351)(213,350)(214,346)(215,348)(216,347)
(217,384)(218,383)(219,382)(220,381)(221,380)(222,379)(223,387)(224,386)
(225,385)(226,394)(227,396)(228,395)(229,391)(230,393)(231,392)(232,388)
(233,390)(234,389)(235,398)(236,397)(237,399)(238,404)(239,403)(240,405)
(241,401)(242,400)(243,402);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(486)!(  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)
( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)
( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)
( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)
( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,167)( 83,168)( 84,166)
( 85,165)( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)
( 93,184)( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)
(101,177)(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)
(109,194)(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)
(117,198)(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)
(125,215)(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)
(133,205)(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)
(141,218)(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)
(149,235)(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)
(157,228)(158,226)(159,227)(160,232)(161,233)(162,234)(247,251)(248,252)
(249,250)(253,262)(254,263)(255,264)(256,269)(257,270)(258,268)(259,267)
(260,265)(261,266)(274,278)(275,279)(276,277)(280,289)(281,290)(282,291)
(283,296)(284,297)(285,295)(286,294)(287,292)(288,293)(301,305)(302,306)
(303,304)(307,316)(308,317)(309,318)(310,323)(311,324)(312,322)(313,321)
(314,319)(315,320)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)
(331,412)(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)
(339,425)(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)
(347,415)(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)
(355,435)(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)
(363,454)(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)
(371,447)(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)
(379,464)(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)
(387,468)(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)
(395,485)(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)
(403,475)(404,476)(405,477);
s1 := Sym(486)!(  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)
(  8, 87)(  9, 86)( 10, 93)( 11, 92)( 12, 91)( 13, 99)( 14, 98)( 15, 97)
( 16, 96)( 17, 95)( 18, 94)( 19,101)( 20,100)( 21,102)( 22,107)( 23,106)
( 24,108)( 25,104)( 26,103)( 27,105)( 28,162)( 29,161)( 30,160)( 31,159)
( 32,158)( 33,157)( 34,156)( 35,155)( 36,154)( 37,143)( 38,142)( 39,144)
( 40,140)( 41,139)( 42,141)( 43,137)( 44,136)( 45,138)( 46,151)( 47,153)
( 48,152)( 49,148)( 50,150)( 51,149)( 52,145)( 53,147)( 54,146)( 55,125)
( 56,124)( 57,126)( 58,122)( 59,121)( 60,123)( 61,119)( 62,118)( 63,120)
( 64,133)( 65,135)( 66,134)( 67,130)( 68,132)( 69,131)( 70,127)( 71,129)
( 72,128)( 73,117)( 74,116)( 75,115)( 76,114)( 77,113)( 78,112)( 79,111)
( 80,110)( 81,109)(163,167)(164,166)(165,168)(169,170)(172,175)(173,177)
(174,176)(179,180)(181,186)(182,185)(183,184)(187,189)(190,235)(191,237)
(192,236)(193,241)(194,243)(195,242)(196,238)(197,240)(198,239)(199,219)
(200,218)(201,217)(202,225)(203,224)(204,223)(205,222)(206,221)(207,220)
(208,227)(209,226)(210,228)(211,233)(212,232)(213,234)(214,230)(215,229)
(216,231)(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)
(251,330)(252,329)(253,336)(254,335)(255,334)(256,342)(257,341)(258,340)
(259,339)(260,338)(261,337)(262,344)(263,343)(264,345)(265,350)(266,349)
(267,351)(268,347)(269,346)(270,348)(271,405)(272,404)(273,403)(274,402)
(275,401)(276,400)(277,399)(278,398)(279,397)(280,386)(281,385)(282,387)
(283,383)(284,382)(285,384)(286,380)(287,379)(288,381)(289,394)(290,396)
(291,395)(292,391)(293,393)(294,392)(295,388)(296,390)(297,389)(298,368)
(299,367)(300,369)(301,365)(302,364)(303,366)(304,362)(305,361)(306,363)
(307,376)(308,378)(309,377)(310,373)(311,375)(312,374)(313,370)(314,372)
(315,371)(316,360)(317,359)(318,358)(319,357)(320,356)(321,355)(322,354)
(323,353)(324,352)(406,410)(407,409)(408,411)(412,413)(415,418)(416,420)
(417,419)(422,423)(424,429)(425,428)(426,427)(430,432)(433,478)(434,480)
(435,479)(436,484)(437,486)(438,485)(439,481)(440,483)(441,482)(442,462)
(443,461)(444,460)(445,468)(446,467)(447,466)(448,465)(449,464)(450,463)
(451,470)(452,469)(453,471)(454,476)(455,475)(456,477)(457,473)(458,472)
(459,474);
s2 := Sym(486)!(  1,271)(  2,273)(  3,272)(  4,277)(  5,279)(  6,278)(  7,274)
(  8,276)(  9,275)( 10,284)( 11,283)( 12,285)( 13,281)( 14,280)( 15,282)
( 16,287)( 17,286)( 18,288)( 19,297)( 20,296)( 21,295)( 22,294)( 23,293)
( 24,292)( 25,291)( 26,290)( 27,289)( 28,244)( 29,246)( 30,245)( 31,250)
( 32,252)( 33,251)( 34,247)( 35,249)( 36,248)( 37,257)( 38,256)( 39,258)
( 40,254)( 41,253)( 42,255)( 43,260)( 44,259)( 45,261)( 46,270)( 47,269)
( 48,268)( 49,267)( 50,266)( 51,265)( 52,264)( 53,263)( 54,262)( 55,299)
( 56,298)( 57,300)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)
( 64,312)( 65,311)( 66,310)( 67,309)( 68,308)( 69,307)( 70,315)( 71,314)
( 72,313)( 73,322)( 74,324)( 75,323)( 76,319)( 77,321)( 78,320)( 79,316)
( 80,318)( 81,317)( 82,437)( 83,436)( 84,438)( 85,434)( 86,433)( 87,435)
( 88,440)( 89,439)( 90,441)( 91,450)( 92,449)( 93,448)( 94,447)( 95,446)
( 96,445)( 97,444)( 98,443)( 99,442)(100,451)(101,453)(102,452)(103,457)
(104,459)(105,458)(106,454)(107,456)(108,455)(109,410)(110,409)(111,411)
(112,407)(113,406)(114,408)(115,413)(116,412)(117,414)(118,423)(119,422)
(120,421)(121,420)(122,419)(123,418)(124,417)(125,416)(126,415)(127,424)
(128,426)(129,425)(130,430)(131,432)(132,431)(133,427)(134,429)(135,428)
(136,465)(137,464)(138,463)(139,462)(140,461)(141,460)(142,468)(143,467)
(144,466)(145,475)(146,477)(147,476)(148,472)(149,474)(150,473)(151,469)
(152,471)(153,470)(154,479)(155,478)(156,480)(157,485)(158,484)(159,486)
(160,482)(161,481)(162,483)(163,356)(164,355)(165,357)(166,353)(167,352)
(168,354)(169,359)(170,358)(171,360)(172,369)(173,368)(174,367)(175,366)
(176,365)(177,364)(178,363)(179,362)(180,361)(181,370)(182,372)(183,371)
(184,376)(185,378)(186,377)(187,373)(188,375)(189,374)(190,329)(191,328)
(192,330)(193,326)(194,325)(195,327)(196,332)(197,331)(198,333)(199,342)
(200,341)(201,340)(202,339)(203,338)(204,337)(205,336)(206,335)(207,334)
(208,343)(209,345)(210,344)(211,349)(212,351)(213,350)(214,346)(215,348)
(216,347)(217,384)(218,383)(219,382)(220,381)(221,380)(222,379)(223,387)
(224,386)(225,385)(226,394)(227,396)(228,395)(229,391)(230,393)(231,392)
(232,388)(233,390)(234,389)(235,398)(236,397)(237,399)(238,404)(239,403)
(240,405)(241,401)(242,400)(243,402);
poly := sub<Sym(486)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1 >;

```
References : None.
to this polytope